10.
$$x ^ { 2 } + 2 x + 3 \equiv ( x + a ) ^ { 2 } + b .$$
- Find the values of the constants \(a\) and \(b\).
- In the space provided below, sketch the graph of \(y = x ^ { 2 } + 2 x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes.
- Find the value of the discriminant of \(x ^ { 2 } + 2 x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b).
The equation \(x ^ { 2 } + k x + 3 = 0\), where \(k\) is a constant, has no real roots.
- Find the set of possible values of \(k\), giving your answer in surd form.